大小分别为 $F\,\mathrm{N}$、$2F\,\mathrm{N}$ 和 $20\,\mathrm{N}$ 的共面力作用于同一点，如图所示。

已知这些力处于平衡状态，求 $F$ 和 $\theta$ 的值。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

Where a numerical value for the acceleration due to gravity ($g$) is needed, use $10\,\mathrm{m\,s^{-2}}$.

需要使用重力加速度 $g$ 的数值时，取 $10\,\mathrm{m\,s^{-2}}$。

先分解竖直、水平平衡，得到20sinθ=2F与20cosθ=√3F，再消去F。

Resolving vertically and horizontally respectively,

Dividing (1) by (2),

Squaring and adding (1) and (2),

Since $F>0$,

三力平衡也可构成闭合力三角形，先余弦定理求F，再正弦定理求θ。

The forces of magnitudes $F$ and $2F$ act at an angle of $60^\circ$ to each other. In the closed triangle of forces, their included interior angle is therefore $120^\circ$.

By the cosine rule,

Thus $F=20/\sqrt7\approx\boxed{7.56}$ to 3 significant figures.

In the force triangle, the angle opposite $F$ is $\theta-30^\circ$. By the sine rule,

Hence

The other two angles of the force triangle sum to $60^\circ$, so $\theta-30^\circ$ is acute. Therefore